# Eureka Math Algebra 1 Common Core End of the year Review ```NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
Name
Date
1. Solve the following equations for 𝑥. Write your answer in set notation.
a.
3𝑥 − 5 = 16
b.
3(𝑥 + 3) − 5 = 16
c.
3(2𝑥 − 3) − 5 = 16
d.
6(𝑥 + 3) − 10 = 32
e.
Which two equations above have the same solution set? Write a sentence explaining how the
properties of equality can be used to determine the pair without having to find the solution set for
each.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
2. Let 𝑐 and 𝑑 be real numbers.
a.
If 𝑐 = 42 + 𝑑 is true, then which is greater: 𝑐 or 𝑑, or are you not able to tell? Explain how you
b.
If 𝑐 = 42 − 𝑑 is true, then which is greater: 𝑐 or 𝑑, or are you not able to tell? Explain how you
3. If 𝑎 &lt; 0 and 𝑐 &gt; 𝑏, circle the expression that is greater:
𝑎(𝑏 − 𝑐)
or
𝑎(𝑐 − 𝑏)
Use the properties of inequalities to explain your choice.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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M1
ALGEBRA I
4. Solve for 𝑥 in each of the equations or inequalities below, and name the property and/or properties used:
a.
3
4
𝑥=9
b.
10 + 3𝑥 = 5𝑥
c.
𝑎+𝑥 = 𝑏
d.
𝑐𝑥 = 𝑑
e.
f.
1
2
𝑥−𝑔&lt;𝑚
𝑞 + 5𝑥 = 7𝑥 − 𝑟
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
g.
h.
3
4
(𝑥 + 2) = 6(𝑥 + 12)
3(5 − 5𝑥) &gt; 5𝑥
5. The equation 3𝑥 + 4 = 5𝑥 − 4 has the solution set {4}.
a.
Explain why the equation (3𝑥 + 4) + 4 = (5𝑥 − 4) + 4 also has the solution set {4}.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
b.
In part (a), the expression (3𝑥 + 4) + 4 is equivalent to the expression 3𝑥 + 8. What is the
definition of equivalent expressions? Why does changing an expression on one side of an equation
to an equivalent expression leave the solution set unchanged?
c.
When we square both sides of the original equation, we get the following new equation:
(3𝑥 + 4)2 = (5𝑥 − 4)2 .
Show that 4 is still a solution to the new equation. Show that 0 is also a solution to the new
equation but is not a solution to the original equation. Write a sentence that describes how the
solution set to an equation may change when both sides of the equation are squared.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
d.
When we replace 𝑥 by 𝑥 2 in the original equation, we get the following new equation:
3𝑥 2 + 4 = 5𝑥 2 − 4.
Use the fact that the solution set to the original equation is {4} to find the solution set to this new
equation.
6. The Zonda Information and Telephone Company (ZI&amp;T) calculates a customer’s total monthly cell phone
charge using the formula,
𝐶 = (𝑏 + 𝑟𝑚)(1 + 𝑡),
where 𝐶 is the total cell phone charge, 𝑏 is a basic monthly fee, 𝑟 is the rate per minute, 𝑚 is the number
of minutes used that month, and 𝑡 is the tax rate.
Solve for 𝑚, the number of minutes the customer used that month.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
7. Students and adults purchased tickets for a recent basketball playoff game. All tickets were sold at the
ticket booth—season passes, discounts, etc., were not allowed.
Student tickets cost \$5 each, and adult tickets cost \$10 each. A total of \$4,500 was collected.
700 tickets were sold.
a.
Write a system of equations that can be used to find the number of student tickets, 𝑠, and the
number of adult tickets, 𝑎, that were sold at the playoff game.
b.
Assuming that the number of students and adults attending would not change, how much more
money could have been collected at the playoff game if the ticket booth charged students and adults
the same price of \$10 per ticket?
c.
Assuming that the number of students and adults attending would not change, how much more
money could have been collected at the playoff game if the student price was kept at \$5 per ticket
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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318
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
8. Alexus is modeling the growth of bacteria for an experiment in science. She assumes that there are
𝐵 bacteria in a Petri dish at 12:00 noon. In reality, each bacterium in the Petri dish subdivides into two
new bacteria approximately every 20 minutes. However, for the purposes of the model, Alexus assumes
that each bacterium subdivides into two new bacteria exactly every 20 minutes.
a.
1
Create a table that shows the total number of bacteria in the Petri dish at hour intervals for 2
hours starting with time 0 to represent 12:00 noon.
3
b.
Write an equation that describes the relationship between total number of bacteria 𝑇 and time ℎ in
hours, assuming there are 𝐵 bacteria in the Petri dish at ℎ = 0.
c.
If Alexus starts with 100 bacteria in the Petri dish, draw a graph that displays the total number of
bacteria with respect to time from 12:00 noon (ℎ = 0) to 4:00 p.m. (ℎ = 4). Label points on your
graph at time ℎ = 0, 1, 2, 3, 4.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
d.
For her experiment, Alexus plans to add an anti-bacterial chemical to the Petri dish at 4:00 p.m. that
is supposed to kill 99.9% of the bacteria instantaneously. If she started with 100 bacteria at
12:00 noon, how many live bacteria might Alexus expect to find in the Petri dish right after she adds
the anti-bacterial chemical?
9. Jack is 27 years older than Susan. In 5 years, he will be 4 times as old as she is.
a.
Find the present ages of Jack and Susan.
b.
What calculations would you do to check if your answer is correct?
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
10.
a.
Find the product: (𝑥 2 − 𝑥 + 1)(2𝑥 2 + 3𝑥 + 2).
b.
Use the results of part (a) to factor 21,112 as a product of a two-digit number and a three-digit
number.
11. Consider the following system of equations with the solution 𝑥 = 3, 𝑦 = 4.
Equation A1:
𝑦 =𝑥+1
Equation A2:
𝑦 = −2𝑥 + 10
a.
Write a unique system of two linear equations with the same solution set. This time make both
linear equations have positive slope.
Equation B1:
________________
Equation B2:
________________
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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M1
ALGEBRA I
b.
The following system of equations was obtained from the original system by adding a multiple of
equation A2 to equation A1.
Equation C1:
𝑦 =𝑥+1
Equation C2:
3𝑦 = −3𝑥 + 21
What multiple of A2 was added to A1?
c.
What is the solution to the system given in part (b)?
d.
For any real number 𝑚, the line 𝑦 = 𝑚(𝑥 − 3) + 4 passes through the point (3,4).
Is it certain, then, that the system of equations
Equation D1:
𝑦 =𝑥+1
Equation D2:
𝑦 = 𝑚(𝑥 − 3) + 4
has only the solution 𝑥 = 3, 𝑦 = 4? Explain.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
12. The local theater in Jamie’s home town has a maximum capacity of 160 people. Jamie shared with Venus
the following graph and said that the shaded region represented all the possible combinations of adult
and child tickets that could be sold for one show.
a.
Venus objected and said there was more than one reason that Jamie’s thinking was flawed. What
reasons could Venus be thinking of?
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
b.
Use equations, inequalities, graphs, and/or words to describe for Jamie the set of all possible
combinations of adult and child tickets that could be sold for one show.
c.
The theater charges \$9 for each adult ticket and \$6 for each child ticket. The theater sold 144
tickets for the first showing of the new release. The total money collected from ticket sales for that
show was \$1,164. Write a system of equations that could be used to find the number of child tickets
and the number of adult tickets sold, and solve the system algebraically. Summarize your findings
using the context of the problem.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
A Progression Toward Mastery
Assessment
1
a–d
A-REI.A.1
e
A-SSE.A.1b
A-REI.B.3
2
a
A-CED.A.3
b
A-CED.A.3
STEP 1
Missing or
and little
evidence of
reasoning or
application of
mathematics to
solve the
problem.
STEP 2
Missing or incorrect
evidence of some
reasoning or
application of
mathematics to
solve the problem.
STEP 3
with some evidence
of reasoning or
application of
mathematics to
solve the problem,
or an incorrect
substantial
evidence of solid
reasoning or
application of
mathematics to
solve the problem.
STEP 4
supported by
substantial evidence
of solid reasoning or
application of
mathematics to
solve the problem.
Student gives a
short incorrect
the question blank.
Student shows at least
one correct step, but
the solution is
incorrect.
Student solves the
equation correctly
(every step that is
shown is correct) but
does not express the
set.
Student solves the
equation correctly
(every step that is
shown is correct) and
as a solution set.
Student does not
incorrectly with
something other
than (b) and (d).
and (d) but does not
demonstrate solid
reasoning in the
explanation.
and (d) and articulates
solid reasoning in the
explanation.
Student responds
incorrectly or leaves
the question blank.
Student responds
correctly that (c) must
be greater but does
not use solid
reasoning to explain
Student responds
correctly that (c) must
be greater but gives
an incomplete or
slightly incorrect
explanation of why.
Student responds
correctly that (c) must
be greater and
supports the statement
with solid,
well-expressed
reasoning.
Student responds
incorrectly or leaves
the question blank.
Student responds
correctly that there is
no way to tell but
does not use solid
reasoning to explain
Student responds
correctly that there is
no way to tell but
gives an incomplete or
slightly incorrect
explanation of why.
Student responds
correctly that there is
no way to tell and
supports the statement
with solid,
well-expressed
reasoning.
Module 1:
and (d) but makes
minor misstatements
in the explanation.
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
3
A-SSE.A.1b
Student responds
incorrectly or leaves
the question blank.
Student responds
correctly by circling
the expression on the
left but does not use
solid reasoning to
Student responds
correctly by circling
the expression on the
left but gives limited
explanation or does
not use the properties
of inequality in the
explanation.
Student responds
correctly by circling the
expression on the left
and gives a complete
explanation that uses
the properties of
inequality.
4
a–h
incorrectly with no
correct steps
shown.
incorrectly but has
one or more correct
steps.
correctly but does not
correctly identify the
property or properties
used.
correctly and correctly
identifies the property
or properties used.
Student does not
demonstrates
incorrect reasoning
throughout.
Student demonstrates
only limited
reasoning.
Student demonstrates
solid reasoning but
falls short of a
makes a minor
misstatement in the
complete and
demonstrates solid
reasoning throughout.
Student does not
demonstrate
understanding of
what the question is
Student makes more
than one
misstatement in the
definition.
Student provides a
mostly correct
definition with a
minor misstatement.
completely and uses a
correct definition
without error or misstatement.
Student makes
mistakes in both
verifications and
demonstrates
incorrect reasoning
or leaves the
question blank.
Student conducts both
verifications but falls
short of articulating
the question.
Student conducts both
verifications and
articulates valid
the question but
makes a minor error in
the verification or a
minor misstatement in
the explanation.
Student conducts both
verifications without
error and articulates
valid reasoning to
incorrectly or does
Student identifies one
or both solutions but
is unable to convey
how the solutions
could be found using
the fact that 4 is a
solution to the original
equation.
Student identifies only
one solution correctly
but articulates the
reasoning of using the
solution to the original
equation to find the
solution to the new
equation.
Student identifies both
solutions correctly and
articulates the
reasoning of using the
solution to the original
equation to find the
solution to the new
equation.
A-REI.A.1
A-REI.B.3
5
a
A-REI.A.1
b
A-REI.A.1
c
A-REI.A.1
d
A-REI.A.1
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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326
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
6
A-CED.A.4
Student does not
evidence of
reasoning.
Student makes more
than one error in the
solution process but
shows some evidence
of reasoning.
valid steps but with
one minor error.
correctly.
7
a–c
Student is unable to
correctly.
part correctly or
shows some evidence
of reasoning in more
than one part.
Student shows solid
evidence of reasoning
in every part but may
make minor errors.
part correctly and
demonstrates and
expresses valid
reasoning throughout.
Student provides no
table or a table with
multiple incorrect
entries.
Student provides a
data table that is
incomplete or has
more than one minor
error.
Student provides a
data table that is
complete but may
have one error or
slightly inaccurate
Student provides a data
table that is complete
and correct with
Student provides no
equation or an
equation that does
not represent
exponential growth.
Student provides an
incorrect equation but
one that models
exponential growth.
Student provides a
form of 𝑇 = 𝐵(2)3ℎ .
Student provides a
form of 𝑇 = 𝐵8ℎ or in
more than one form,
such as 𝑇 = 𝐵(2)3ℎ
and 𝑇 = 𝐵8ℎ .
Student provides no
graph or a grossly
inaccurate graph.
Student provides a
graph with an
inaccurate shape but
provides some
evidence of reasoning
in labeling the axes
and/or data points.
Student creates a
graph with correct
general shape but may
leave off or make an
error on one or two
axes or data points.
Student creates a
complete graph with
correctly labeled axes
and correctly labeled
data points (or a data
table) showing the
values for
ℎ = 0, 1, 2, 3, 4.
Student provides no
with no evidence of
reasoning in arriving
Student provides
limited evidence of
reasoning and an
409.6 bacteria are
alive.
bacteria are alive.
A-CED.A.3
8
a
A-CED.A.2
b
A-CED.A.2
c
A-CED.A.2
d
A-CED.A.2
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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This file derived from ALG I-M1-TE-1.3.0-07.2015
327
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
9
a
A-CED.A.1
b
A-REI.B.3
10
a–b
A-APR.A.1
11
a
A-REI.C.6
b
A-REI.C.6
Student writes
incorrect equations
or does not provide
equations.
incorrect, but at least
one of the equations
is correct. Student
makes a gross error in
the solution, makes
more than one minor
error in the solution
process, or has one of
the two equations
incorrect.
Both equations are
correct, but student
makes a minor
mistake in finding the
solution.
Both equations are
correct and student
solves them correctly
that Jack is 31 and
Susan is 4.
Student does not
completely
Student articulates
only one of the
calculations correctly.
Student articulates
the two calculations
but with a minor misstatement in one of
the descriptions.
Student articulates
both calculations
correctly.
Student work is
blank or
demonstrates no
understanding of
multiplication of
polynomials, nor
how to apply
part (a) to arrive at
part (b).
Student makes more
than one error in the
multiplication but
demonstrates some
understanding of
multiplication of
polynomials. Student
may not be able to
garner or apply
information from
part (a) to use in
correctly.
Student demonstrates
the ability to multiply
the polynomials
(expressing the
product as a sum of
monomials with like
terms combined) and
to apply the structure
from part (a) to solve
part (b). There may
be minor errors.
Student demonstrates
the ability to multiply
the polynomials
(expressing the product
as a sum of monomials
with like terms
combined) and to apply
the structure from
part (a) to solve part (b)
as 91(232).
Student is unable to
demonstrate the
understanding that
two equations with
(3, 4) as a solution
are needed.
Student provides two
equations that have
(3, 4) as a solution (or
attempts to provide
such equations) but
makes one or more
errors. Student may
provide an equation
with a negative slope.
Student shows one
minor error in the
to provide two
equations both
containing (3, 4) as a
solution and both with
positive slope.
Student provides two
equations both
containing (3, 4) as a
solution and both with
positive slope.
Student is unable to
identify the multiple
correctly.
Student identifies the
multiple as 3.
N/A
Student correctly
identifies the multiple
as 2.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
c
A-REI.C.6
d
A-REI.C.5
A-REI.C.6
A-REI.D.10
12
a
MP.2
A-REI.D.12
b
A-CED.A.2
A-REI.D.10
A-REI.D.12
Student is unable to
demonstrate even a
partial
understanding of
how to find the
solution to the
system.
Student shows some
reasoning required to
find the solution but
makes multiple errors.
Student makes a
minor error in finding
the solution point.
Student successfully
identifies the solution
point as (3, 4).
Student is unable to
with any solid
reasoning.
Student concludes yes
or no but is only able
to express limited
reasoning in support
Student correctly
explains that all the
systems have the
solution point (3, 4)
but incorrectly
assumes this is true
for all cases of 𝑚.
Student correctly
explains that while in
most cases this is true,
if 𝑚 = 1, the two lines
are coinciding lines,
resulting in a solution
set consisting of all the
points on the line.
Student is unable to
articulate any sound
reasons.
Student is only able to
articulate one sound
reason.
Student provides two
sound reasons but
makes minor errors in
the expression of
reasoning.
Student is able to
articulate at least two
valid reasons. Valid
reasons include the
following: the graph
assumes 𝑥 could be less
than zero, the graph
assumes 𝑦 could be less
than zero, the graph
assumes 𝑎 and 𝑏 could
be non-whole numbers,
the graph assumes 160
children could attend
Student is unable to
communicate a
relevant
requirement of the
solution set.
Student provides a
verbal description that
lacks precision and
accuracy but
demonstrates some
solution within the
context of the
problem.
Student makes minor
errors in
communicating the
idea that both (a) and
(b) must be whole
numbers whose sum
is less than or equal to
160.
Student communicates
effectively that both (a)
and (b) must be whole
numbers whose sum is
less than or equal to
160.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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This file derived from ALG I-M1-TE-1.3.0-07.2015
329
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
c
A-CED.A.2
A-REI.C.6
Student is unable to
demonstrate any
substantive
understanding in
how to create the
equations and solve
the system of
equations.
Module 1:
Student makes
multiple errors in the
equations and/or
solving process but
demonstrates some
understanding of how
to create equations to
represent a context
and/or solve the
system of equations.
Student makes minor
errors in the
equations but solves
the system accurately,
or the student creates
the correct equations
but makes a minor
error in solving the
system of equations.
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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Student correctly
writes the equations to
represent the system.
Student solves the
system accurately and
summarizes by defining
or describing the values
of the variable in the
context of the problem
(i.e., that there are
44 child tickets sold.)
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Name
Date
1. Solve the following equations for 𝑥. Write your answer in set notation.
a.
3𝑥 − 5 = 16
3x = 21
Solution set: {7}
x=7
b.
3(𝑥 + 3) − 5 = 16
3x + 9 - 5 = 16
Solution set: {4}
3x = 12
x=4
c.
3(2𝑥 − 3) − 5 = 16
6x - 9 - 5 = 16
Solution set: {5}
6x - 14 = 16
6x = 30
x=5
d.
6(𝑥 + 3) − 10 = 32
6x + 18 - 10 = 32
Solution set: {4}
6x = 24
x=4
e.
Which two equations above have the same solution set? Write a sentence explaining how the
properties of equality can be used to determine the pair without having to find the solution set for
each.
Problems (b) and (d) have the same solution set. The expressions on each side of the
equal sign for (d) are twice those for (b). So, if (left side) = (right side) is true for only
some x-values, then 2(left side) = 2(right side) will be true for exactly the same xvalues. Or simply, applying the multiplicative property of equality does not change the
solution set.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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ALGEBRA I
2. Let 𝑐 and 𝑑 be real numbers.
a.
If 𝑐 = 42 + 𝑑 is true, then which is greater: 𝑐 or 𝑑 or are you not able to tell? Explain how you know
c must be greater because c is always 42 more than d.
b.
If 𝑐 = 42 − 𝑑 is true, then which is greater: 𝑐 or 𝑑 or are you not able to tell? Explain how you know
There is no way to tell. We only know that the sum of c and d is 42. If d were 10, c
would be 32 and, therefore, greater than d. But if d were 40, c would be 2 and,
therefore, less than d.
3. If 𝑎 &lt; 0 and 𝑐 &gt; 𝑏, circle the expression that is greater:
𝑎(𝑏 − 𝑐)
or 𝑎(𝑐 − 𝑏)
Use the properties of inequalities to explain your choice.
Since c &gt; b,
Since c &gt; b,
it follows that 0 &gt; b − c,
it follows that c − b &gt; 0.
and since a &lt; 0, a is negative,
so (c − b) is positive. And since a is
and the product of two negatives will be
negative, the product of
a positive.
a ∙ (c − b) &lt; a ∙ (b − c).
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
4. Solve for 𝑥 in each of the equations or inequalities below and name the property and/or properties used:
a.
3
4
𝑥=9
4
x=9∙ ( )
3
x = 12
b.
Multiplication property of equality
10 + 3𝑥 = 5𝑥
10 = 2x
5=x
c.
Multiplication property of equality
𝑎+𝑥 = 𝑏
x=b−a
d.
𝑐𝑥 = 𝑑
x=
e.
1
2
d
, c≠0
c
𝑥−𝑔&lt;𝑚
1
x&lt;m+g
2
x &lt; 2 ⋅ (m + g)
f.
Multiplication property of equality
Multiplication property of equality
𝑞 + 5𝑥 = 7𝑥 − 𝑟
q + r = 2x
(q+r)
=x
2
Multiplication property of equality
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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M1
ALGEBRA I
g.
3
4
(𝑥 + 2) = 6(𝑥 + 12)
3 ∙ (x + 2) = 24 ∙ (x + 12)
3x + 6 = 24x + 288
Distributive property
−
Addition property of equality and multiplication
282
=x
21
94
−
=x
7
94
−
=x
7
h.
Multiplication property of equality
Property of equality
3(5 – 5𝑥) &gt; 5𝑥
15 − 15x &gt; 5x
15 &gt; 20x
Distributive property
3
&gt;x
4
Multiplication property of equality
5. The equation, 3𝑥 + 4 = 5𝑥 − 4, has the solution set {4}.
a.
Explain why the equation, (3𝑥 + 4) + 4 = (5𝑥 − 4) + 4, also has the solution set {4}.
Since the new equation can be created by applying the addition property of equality,
the solution set does not change.
OR
Each side of this equation is 4 more than the sides of the original equation. Whatever
value(s) make 3x + 4 = 5x − 4 true would also make 4 more than 3x + 4 equal to 4
more than 5x − 4.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
b.
In part (a), the expression (3𝑥 + 4) + 4 is equivalent to the expression 3𝑥 + 8. What is the
definition of equivalent algebraic expressions? Describe why changing an expression on one side of
an equation to an equivalent expression leaves the solution set unchanged?
Algebraic expressions are equivalent if (possibly repeated) use of the distributive,
associative, and commutative properties and/or the properties of rational exponents can
be applied to one expression to convert it to the other expression.
When two expressions are equivalent, assigning the same value to x in both expressions
will give an equivalent numerical expression, which then evaluates to the same number.
Therefore, changing the expression to something equivalent will not change the truth
value of the equation once values are assigned to x.
c.
When we square both sides of the original equation, we get the following new equation:
(3𝑥 + 4)2 = (5𝑥 − 4)2 .
Show that 4 is still a solution to the new equation. Show that 0 is also a solution to the new
equation but is not a solution to the original equation. Write a sentence that describes how the
solution set to an equation may change when both sides of the equation are squared.
(3 ∙ 4 + 4)2 = (5 ∙ 4 − 4)2 gives 162 = 166 , which is true.
(3 ∙ 0 + 4)2 =(5 ∙ 0 − 4)2 gives 42 =(−4)2 , which is true.
But, (3 ∙ 0 + 4) = (5 ∙ 0 − 4) gives 4 = − 4, which is false.
When both sides are squared, you might introduce new numbers to the solution set
because statements like 4 = − 4 are false, but statements like 42 = (−4)2 are true.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
d.
When we replace 𝑥 by 𝑥 2 in the original equation, we get the following new equation:
3𝑥 2 + 4 = 5𝑥 2 − 4.
Use the fact that the solution set to the original equation is {4} to find the solution set to this new
equation.
Since the original equation 3x + 4 = 5x − 4 was true when x = 4, the new equation
3x2 + 4 = 5x2 − 4 should be true when x2 = 4. And, x2 = 4 when x = 2, so the solution set
to the new equation is {−2,2}.
6. The Zonda Information and Telephone Company calculates a customer’s total monthly cell phone charge
using the formula,
𝐶 = (𝑏 + 𝑟𝑚)(1 + 𝑡),
where 𝐶 is the total cell phone charge, 𝑏 is a basic monthly fee, 𝑟 is the rate per minute, 𝑚 is the number
of minutes used that month, and 𝑡 is the tax rate.
Solve for 𝑚, the number of minutes the customer used that month.
C = b + bt + rm + rmt
C − b − bt = m ∙ (r + rt)
C − b − bt
=m
r + rt
Module 1:
t≠ −1
r≠0
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
7. Students and adults purchased tickets for a recent basketball playoff game. All tickets were sold at the
ticket booth—season passes, discounts, etc., were not allowed.
Student tickets cost \$5 each, and adult tickets cost \$10 each. A total of \$4,500 was collected.
700 tickets were sold.
a.
Write a system of equations that can be used to find the number of student tickets, 𝑠, and the
number of adult tickets, 𝑎, that were sold at the playoff game.
5s + 10a = 4500
s + a = 700
b.
Assuming that the number of students and adults attending would not change, how much more
money could have been collected at the playoff game if the ticket booth charged students and adults
the same price of \$10 per ticket?
700 &times; \$10 = \$7000
\$7000 − \$4500 = \$2500 more
c.
Assuming that the number of students and adults attending would not change, how much more
money could have been collected at the playoff game if the student price was kept at \$5 per ticket
First solve for a and s
5s + 10a = 4500
-5s − 5a = − 3500
5a = 1000
a = 200
s = 500
\$5 ∙ (500) + \$15 ∙ (200) = \$5500
\$1,000 more
OR
\$5 more per adult ticket (200 ∙ \$5 = \$1000 more)
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
8. Alexus is modeling the growth of bacteria for an experiment in science. She assumes that there are 𝐵
bacteria in a Petri dish at 12:00 noon. In reality, each bacterium in the Petri dish subdivides into two new
bacteria approximately every 20 minutes. However, for the purposes of the model, Alexus assumes that
each bacterium subdivides into two new bacteria exactly every 20 minutes.
a.
1
Create a table that shows the total number of bacteria in the Petri dish at hour intervals for
3
2 hours starting with time 0 to represent 12:00 noon.
b.
Time
Number of Bacteria
0
B
1/3 hour
2B
2/3 hour
4B
1 hour
8B
1 1/3 hour
16B
1 2/3 hour
32B
2 hour
64B
Write an equation that describes the relationship between total number of bacteria T and time h in
hours, assuming there are 𝐵 bacteria in the Petri dish at ℎ = 0.
T = B ∙ (2)3h or T = B ∙ 8h
c.
If Alexus starts with 100 bacteria in the Petri dish, draw a graph that displays the total number of
bacteria with respect to time from 12:00 noon (ℎ = 0) to 4:00 p.m. (ℎ = 4). Label points on your
graph at time ℎ = 0, 1, 2, 3, 4.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
d.
For her experiment, Alexus plans to add an anti-bacterial chemical to the Petri dish at 4:00 p.m. that
is supposed to kill 99.9% of the bacteria instantaneously. If she started with 100 bacteria at
12:00 noon, how many live bacteria might Alexus expect to find in the Petri dish right after she adds
the anti-bacterial chemical?
(1 − 0.999) ∙ 409 600 = 409.6
9. Jack is 27 years older than Susan. In 5 years time, he will be 4 times as old as she is.
a.
Find the present ages of Jack and Susan.
J = S + 27
J + 5 = 4 ∙ (S + 5)
S + 27 + 5 = 4S + 20
S + 32 = 4S + 20
12 = 3S
S=4
J = 4 + 27
J = 31
Jack is 31 and Susan is 4.
b.
What calculations would you do to check if your answer is correct?
Is Jack’s age – Susan’s age = 27?
Add 5 years to Jack’s and Susan’s ages, and see if that makes Jack 4 times as old as
Susan.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
10.
a.
Find the product: (𝑥 2 − 𝑥 + 1)(2𝑥 2 + 3𝑥 + 2)
2x4 + 3x3 + 2x2 − 2x3 − 3x2 − 2x + 2x2 + 3x + 2
2x4 + x3 + x2 + x + 2
b.
Use the results of part (a) to factor 21,112 as a product of a two-digit number and a three-digit
number.
(100 − 10 + 1) ∙ (200 + 30 + 2)
(91) ∙ (232)
11. Consider the following system of equations with the solution 𝑥 = 3, 𝑦 = 4.
Equation A1:
𝑦 =𝑥+1
Equation A2:
𝑦 = −2𝑥 + 10
a.
Write a unique system of two linear equations with the same solution set. This time make both
linear equations have positive slope.
Equation B1:
Equation B2:
Module 1:
4
y = _______
x
3
y=x+1
________
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
b.
The following system of equations was obtained from the original system by adding a multiple of
equation A2 to equation A1.
Equation C1:
𝑦 =𝑥+1
Equation C2:
3𝑦 = −3𝑥 + 21
What multiple of A2 was added to A1?
2 times A2 was added to A1.
c.
What is the solution to the system given in part (b)?
(3,4)
d.
For any real number 𝑚, the line 𝑦 = 𝑚(𝑥 − 3) + 4 passes through the point (3,4).
Is it certain then that the system of equations:
Equation D1:
𝑦 =𝑥+1
Equation D2:
𝑦 = 𝑚(𝑥 − 3) + 4
has only the solution 𝑥 = 3, 𝑦 = 4? Explain.
No. If m = 1, then the two lines have the same slope. Both lines pass through the point
(3,4), and the lines are parallel; therefore, they coincide. There are infinite solutions.
The solution set is all the points on the line. Any other nonzero value of m would create
a system with the only solution of (3,4).
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
12. The local theater in Jamie’s home town has a maximum capacity of 160 people. Jamie shared with Venus
the following graph and said that the shaded region represented all the possible combinations of adult
and child tickets that could be sold for one show.
a.
Venus objected and said there was more than one reason that Jamie’s thinking was flawed. What
reasons could Venus be thinking of?
1. The graph implies that the number of tickets sold could be a fractional amount, but
really it only makes sense to sell whole number tickets. x and y must be whole
numbers.
2. The graph also shows that negative ticket amounts could be sold, which does not
make sense.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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M1
ALGEBRA I
b.
Use equations, inequalities, graphs, and/or words to describe for Jamie the set of all possible
combinations of adult and child tickets that could be sold for one show.
The system would be {
c.
a + c ≤ 160
a ≥ 0 where a and c are whole numbers.
c≥0
The theater charges \$9 for each adult ticket and \$6 for each child ticket. The theater sold 144
tickets for the first showing of the new release. The total money collected from ticket sales for that
show was \$1,164. Write a system of equations that could be used to find the number of child tickets
and the number of adult tickets sold, and solve the system algebraically. Summarize your findings
using the context of the problem.
a: the number of adult tickets sold (must be a whole number)
c: the number of child tickets sold (must be a whole number)
9a + 6c = 1164
{
a + c = 144
9a + 6c = 1164
-6a − 6c = -864
3a = 300
a = 100, c = 44
In all, 100 adult tickets and 44 child tickets were sold.
Module 1:
Relationships Between Quantities and Reasoning with Equations and
Their Graphs
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343